A356715 Total number of distinct numbers that can be obtained by starting with 1 and applying the "Choix de Bruxelles", version 2 operation at most n times in ternary (base 3).

1, 2, 3, 6, 11, 26, 68, 177, 492, 1403, 4113, 12149, 36225, 108268, 324529, 973163, 2920533, 8764041, 26303715, 78935398, 236878491, 710783343

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Eric Angelini, Lars Blomberg, Charlie Neder, Remy Sigrist, and N. J. A. Sloane, "Choix de Bruxelles": A New Operation on Positive Integers, arXiv:1902.01444 [math.NT], Feb 2019; Fib. Quart. 57:3 (2019), 195-200.

J. Conrad, Python program.

Example

For n = 2, a(2) = 3 since the numbers obtained are (in base 3): 1, 2, 11.
For n = 4, they expand to a(5) = 11 numbers (in base 3): 1, 2, 11, 12, 21, 22, 101, 111, 112, 121, 211.

Prog

(Python) See links
(Python)
from itertools import islice
from sympy.ntheory import digits
def fd(d, b): return sum(b**i*di for i, di in enumerate(d[::-1]))
def cdb2(n, base=3):
    d, out = digits(n, base)[1:], {n}
    for l in range(1, len(d)+1):
        for i in range(len(d)+1-l):
            if d[i] == 0: continue
            t = fd(d[i:i+l], base)
            out.add(fd(d[:i] + digits(2*t, base)[1:] + d[i+l:], base))
            if t&1 == 0:
                out.add(fd(d[:i] + digits(t//2, base)[1:] + d[i+l:], base))
    return out
def agen():
    reach, expand = {1}, [1]
    while True:
        yield len(reach) #; print(reach); print([digits(t, 3)[1:] for t in sorted(reach)])
        newreach = {r for q in expand for r in cdb2(q) if r not in reach}
        reach |= newreach
        expand = list(newreach)
print(list(islice(agen(), 10))) # Michael S. Branicky, Aug 24 2022

Crossrefs

Cf. A323289 (decimal), A356511 (base 12)

Sequence in context: A094927 A024423 A115114 * A324768 A086209 A022490

Adjacent sequences: A356712 A356713 A356714 * A356716 A356717 A356718

Keyword

nonn,base,more

Author

J. Conrad, Aug 24 2022

Extensions

a(15)-a(19) from Michael S. Branicky, Aug 24 2022

a(20)-a(21) from Michael S. Branicky, Aug 30 2022

Status

approved